I might be making some naive assumptions here, but I have a hard time seeing what the purpose of congregations are in the context of priesthood of all believers.

As I understand it, priesthood of all believers basically says that all people have direct and equal access to God, as well as equal privilege and authority to minister. What then is the purpose of congregating? Why are those congregations well organized?

That's a good question. Catholics believe baptism makes you priest, prophet and king as well as in infant baptism. So, perhaps the assumption you're making is that all grace is the same grace.
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Peter TurnerNov 29 '11 at 17:22

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Congregational worship is not about "access to God." It's about living in community with other believers. If it was only about "access to God," then even Catholic and Jewish worship (which do use priests for some form of "access to God") would be quite different.
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FlimzyNov 29 '11 at 19:30

@Peter: that might be the case, but it's a common share in the priesthood of Christ which isn't the same as Holy Orders which sets a man apart from the Faithful in order to offer sacrifice and confect the Sacraments on their behalf. I'm not asserting you don't know this, just stating it for non-Catholics who might not get the "common Priesthood" thing as understood in Catholic teaching.
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Audio SanctoFeb 3 '12 at 15:18

3 Answers
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At least part of this answer I believe comes through obedience to the many ‘one another’ commands of the New Testament.

We are exhorted in various ways to be involved with and caring for one another:

To admonish one another (Rom. 15:14)

To comfort and encourage one another (1 Thess. 4:18; 5:11; Heb. 3:13)

To worship with one another (Eph. 5:19; Col. 3:16; Heb. 10:25)

To bear one another’s burdens (Gal. 6:2)

To always seek the good of one another (1 Thess. 5:15)

To be honest with one another (Col. 3:9)

To show hospitality to one another (1 Pet. 4:9)

To be at peace with one another (Mark 9:50)

An overview of the Greek 'one another' is given in Christian Fellowship by J. Hampton Keathley, III which states:

The expression ‘one another’ is a translation of a reciprocal pronoun
in the Greek New Testament. Reciprocal means mutual, shared, shown or
felt alike by both sides, united in feelings, actions,
responsibilities, and attitudes. Synonyms include: common, mutual,
fellowship, and shared—ideas that are at the heart of the doctrine of
fellowship. In usage, this pronoun is used in statements and
injunctions to believers regarding shared and mutual responsibilities.
In emphasis, it focuses us on our need of the ministry and aid of
others, of our duty to care for others as partners in the body of
Christ, and of how we can experience true fellowship. Therefore, a
study of the ‘one another’ commands of Scripture would be tremendously
helpful in the matter of New Testament fellowship. For a detailed
study of the doctrine, see the study on the “One Another” Commands of
Scripture..

I believe it would be quite hard, if not impossible, to follow these instructions without some form of Congregationalism.

The purpose of congregational gathering is not exclusively to provide the laity access to someone who can be a mediator between them and God, so the assumption that the priesthood of all believers is mutually exclusive is invalid.

Gathering as a congregation has many purposes, including mutual edification, encouragement toward love and good deeds, instruction in the word, serving one another, community, etc.

So, the biblical idea that we all have one Mediator between us and God in no way precludes the call to fellowship with one another.

You are correct in viewing the priesthood of all believers as obviating the need for a mediator between oneself and God - but there is also incumbent upon the participant a duty, as a priest, to minister to one's fellow congregants.

Put simply, without a congregation, there is no one to minister.

As a member in the priesthood of all believers, I am to minister to others in my church, and my church ministers to me. We sharpen, encourage, and build up "the brethren," meaning that we all serve as priests for one another, without respect to hierarchy.